On the dependence of the first exit times on the fluctuations of the domain boundary*

The paper studies first exit times from domains for diffusion processes and their dependence on variations of the boundary. We establish some robustness of the first exit times with respect to the fluctuations of the boundary. More precisely, we present an estimate of the L1-distance between exit times from two regions via expectations of exit times.


Introduction
This paper studies path-wise dependence on fluctuations of the boundary for first exit times of diffusion processes.It is known that first exit times from a region for smooth solutions of ordinary equations do not depend continuously on variations of the initial data or on the boundary of the region.On the other hand, first exit times for non-smooth trajectories of diffusion processes have some path-wise regularity with respect to these variations; some results can be found in [1,2].In this short note, we present an effective estimate of L 1 -distance between exit times from two regions for a diffusion process via expectations of exit times.This means that we established some robustness of the first exit times with respect to the fluctuations of the boundary.This is an extension of the result obtained in [1,2], where first exits were considered from the fixed domain that was the same for both processes.

The result
Let (Ω, F, P) be a standard probability space, where Ω is a set of elementary events, F is a complete σ-algebra of events, and P is a probability measure.Let w(t) be a standard d-dimensional Wiener process.Consider a diffusion process y(t) with the values in R n such that dy(t) = f (y(t))dt + β(y(t))dw(t), t > 0, y(0) = a.Here f : R n → R n and β : R n → R b×d are measurable functions, a is a random vector with values in R n that is independent on w(•).We assume that all the components of the functions f , β are continuously differentiable, and that β(x)β(x) ≥ cI n for some c > 0.
Here I n is the unit matrix in R n×n .
For x ∈ R n , we denote by y x (t) the solution of (1) with the initial condition y(0) = x.
For a set Γ ∈ R n , we denote τ (Γ) ∆ = inf{t : y(t) ∈ Γ}.This is the first times of achieving Γ for y(t).Similarly, we denote τ x (Γ) We assume that the boundaries of D i are C 1 -smooth.
Note the theorem is oriented on the case where Theorem 2.1 establishes some robustness in L 1 -metric of the first exit times with respect to the fluctuations of the boundary, since the expectations in the right hand part of (2.2) are supposed to be small if Γ 1 ≈ Γ 2 , in a typical case.This can be illustrated as the following.
and ε ≥ 0. This can be easily found from the corresponding problems (2.2), (2.5) below that have a trivial quadratic polynomial solution in this case.We have that D0 Proof of Theorem 2.1.Let {F t } t≥0 be the filtration generated by w(t) and a.Let e 1 and e 2 be the indicator functions of the random events {τ (Γ 1 ) > τ (Γ 2 )} and {τ (Γ 2 ) > τ (Γ 1 )} respectively.
Let τ ).The random variables e i are measurable with respect to the σ-algebras F τ and F τ (Γi) , i = 1, 2, associated with the Markov times τ and τ (Γ i ) (Markov times with respect to the filtration {F t }); see, e.g., [3], Section 4.2. Let Here the differential operator where f k , y l , and b k,l are the components of the vectors f , y and the matrix b = ββ .By Theorem 2.2 from [1] Similarly, replacing the indices 1, 2 in (2.4) by 2, 1, we obtain that Now the assertion of Theorem 2.1 follows. 2 Remark 2.3.We have assumed that the boundaries and coefficients are smooth, the diffusion is non-degenerate, and the domains are bounded.In fact, these conditions can be lifted provided that the right hand part of (2.2) is finite.In particular, a similar result can be obtained for first exit times of a degenerate diffusion process (y(t), t) from cylindrical domains D i × (0, T ), i = 1, 2, T > 0; in this case, coefficients f and β can be time dependent.